Secretary Problems with Convex Costs

12/2011; DOI: 10.1007/978-3-642-31594-7_7
Source: arXiv


We consider online resource allocation problems where given a set of requests
our goal is to select a subset that maximizes a value minus cost type of
objective function. Requests are presented online in random order, and each
request possesses an adversarial value and an adversarial size. The online
algorithm must make an irrevocable accept/reject decision as soon as it sees
each request. The "profit" of a set of accepted requests is its total value
minus a convex cost function of its total size. This problem falls within the
framework of secretary problems. Unlike previous work in that area, one of the
main challenges we face is that the objective function can be positive or
negative and we must guard against accepting requests that look good early on
but cause the solution to have an arbitrarily large cost as more requests are
accepted. This requires designing new techniques.
We study this problem under various feasibility constraints and present
online algorithms with competitive ratios only a constant factor worse than
those known in the absence of costs for the same feasibility constraints. We
also consider a multi-dimensional version of the problem that generalizes
multi-dimensional knapsack within a secretary framework. In the absence of any
feasibility constraints, we present an O(l) competitive algorithm where l is
the number of dimensions; this matches within constant factors the best known
ratio for multi-dimensional knapsack secretary.

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    • "Recently, increased interest arose in nonlinear versions of the secretary problem, with a focus on the maximization of a non-negative submodular function 2 [5] [6] [14] [23] [32], leading to the submodular secretary problem. Submodular functions have widespread use as valuation functions because they reflect the property of diminishing returns, i.e., the marginal value of an element is the bigger the fewer elements have been selected so far. "
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    ABSTRACT: The secretary problems are of types of the optimal stopping theory problems and are also distinguished as well known problems in applied probability, statistics and decision theory fields. The importance of these kinds of problems defines many new conditions which have been discussed till now. The significance of these types of problems in social issues causes to define many conditions which are vastly discussed. In this paper, a new condition of this problem is considered. These conditions which have been performed using a real time method are based on Multi-Agent Systems techniques. In this paper, after evaluating this method, the resulted answers are examined using Multi-Agent system techniques.
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